Power-law distributions from additive preferential redistributions

Area of Science:

• Statistical mechanics
• Complex systems

Background:

• Power-law distributions (e.g., Zipf's law) are common in nature but often explained by growth models.
• Existing models may not apply when growth is absent or unexpected.

Purpose of the Study:

• Introduce a nongrowth model to generate power-law distributions.
• Explore an alternative mechanism based on conserved quantities and interactions.

Main Methods:

• Developed a mathematical model with elements possessing quantities undergoing redistribution via binary random interactions.
• Applied a simple additive preferential rule while conserving the total quantity.
• Performed analytical and numerical analyses to obtain stationary distributions and identify scaling behavior.

Main Results:

• The nongrowth model successfully generates power-law distributions with a Zipf exponent.
• Stationary distributions were obtained analytically and numerically across various model parameters.
• Scaling behavior was observed within specific parameter ranges, demonstrating the model's effectiveness.

Conclusions:

• This nongrowth model provides a viable mechanism for power-law distributions without relying on growth processes.
• The model highlights the role of conserved quantities and preferential interactions in generating scale-free phenomena.
• Applicable to diverse systems including personal wealth, city sizes, and scale-free network generation via rewiring.